In this paper, we propose a seamless adaptive phase II/III design based on binary endpoints for evaluating the efficacy of drugs. One intriguing feature for such designs is the reduction in development time. Since the phase II and phase III trials are combined into a single trial, there will be no lead time between the phase II and phase III clinical developments. Also the data collected from the phase II stage will also be included in the analysis of the phase III stage, and thus the sample size reduction is possible.
As we mentioned, the success rate of drug development has been declined drastically in recent years. Some possible reasons may be that the patient populations recruited for the phase II and phase III trials are different, and also schemes used at the phase II and phase III trials are different. However, in our design, the phase II and phase III trials are conducted in the same protocol with the same inclusion/exclusion criteria, the same study design, the same control group, the same methods for evaluation, and the same efficacy/safety endpoints, the above issues concerning the traditional clinical development can therefore be avoided.
The selection of C1 may be another important issue. As we know, larger value of C1 will produce higher probability of early stopping, and thus reduce the expected total sample sizes. On the other hand, larger value of C1 can also increase the success probability of the phase III stage for the clinical development. However, most of all, the determination of C1 should meet the minimal clinically meaningful requirement that an investigator would need to observe before continuing accrual onto the phase
In the traditional approach which the phase II trial and the phase III trial are conducted separately, the overall type I error rate (alpha error) is in fact
0025 . 0 05 . 0 05 .
0 if the type I error rate at each stage is 0.05. In the proposed adaptive seamless phase II/III design, the total type I error rate is equal to 0.05. Thus, the type I error rate of the proposed design is 20 times larger than that of the traditional approach. Consequently, the traditional approach seems too conservative.
On the other hand, if the power at the phase II or III stage equals 0.8, then the overall power rate of the traditional approach is 0.80.80.64. Nevertheless, the overall power of our design is 0.8 which is 1.25 times larger than that of the traditional design.
That is, our phase II/III design is much powerful than the traditional approach. This phenomenon can be observed from Figures 1, 2, 3 and 4.
We may get much benefit from the adaptive seamless phase II/III design, but there is something we should pay attention to. This design is not applicable to all clinical trials; we should weigh gains and losses of it. The most important feasibility consideration that Maca et al. (2006) mentioned is the amount of time for a patient required to be followed to reach the endpoint for the selection decision. There is a period called “transition” period in which patients have been randomized but have not been followed long enough to get the endpoint for the selection prior to the interim analysis. If the time needed to follow up is relative short than the enrollment time, the enrollment of few patients should not be interrupted during this period. Even though some patients might be assigned to the treatment that might not continue for the confirmatory stage, they might provide information about safety. Alternatively, if the endpoint duration is too long, the enrollment might be temporarily broke off to avoid randomization of many patients which resulted in unacceptable inefficiencies. In this case, however, the trial would be interrupted and it might erode the time savings of an
adaptive seamless design. Maca et al. also recommend using well-established and well-understood endpoints or surrogate markers when adopting this design. In spite of the same endpoint of the two stages in this proposed design, treatment selection of a general adaptive seamless design may be based on different endpoints. For instance, the learning stage adopts a short-term or surrogate endpoint, whereas the confirmatory stage uses a long-term endpoint. The design is not feasible if we have to determine the endpoint of the phase II stage in a new disease area to be used in the next stage.
Before we decide to use an adaptive seamless design, we should check whether the method is feasible or not. We all hope that such a design could provide great benefit to patients, industry, and academia.
Tables
Table 1. Designs for ri r0 0.20, K 1,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.25 0 0.22 0.08 11 26 47.53 40 40 0.46
0.05 0.18 0.08 13 26 39.47 40 40 0.49
0.1 0.3 0 0.27 0.10 14 35 62.32 50 50 0.49 0.05 0.21 0.09 18 36 57.04 50 50 0.54
0.2 0.4 0 0.30 0.11 19 49 86.06 65 65 0.52
0.05 0.25 0.11 24 50 79.64 65 65 0.57
0.3 0.5 0 0.32 0.12 22 56 98.94 75 75 0.52
0.05 0.26 0.12 28 57 93.19 75 75 0.57
0.4 0.6 0 0.34 0.13 23 59 103.88 78 78 0.53 0.05 0.28 0.12 29 60 97.99 78 78 0.57
0.5 0.7 0 0.35 0.13 23 56 101.06 75 75 0.53 0.05 0.29 0.13 28 58 95.32 75 75 0.57
0.6 0.8 0 0.37 0.14 20 49 88.18 65 65 0.53
0.05 0.30 0.13 25 50 84.51 65 65 0.58
0.7 0.9 0 0.40 0.15 15 37 66.37 50 50 0.52
0.05 0.33 0.14 19 37 64.27 50 50 0.58
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly.
n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 2. Designs for ri r0 0.15, K 1,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.2 0 0.17 0.07 18 42 77.31 61 61 0.49
0.05 0.13 0.06 22 45 61.82 61 61 0.55
0.1 0.25 0 0.21 0.08 24 59 105.97 80 80 0.52
0.05 0.15 0.07 31 63 91.47 80 80 0.59
0.2 0.35 0 0.23 0.09 33 83 147.43 110 110 0.53 0.05 0.17 0.08 45 89 135.75 110 110 0.61
0.3 0.45 0 0.24 0.09 40 99 177.26 129 129 0.54 0.05 0.18 0.08 53 105 162.22 129 129 0.61
0.4 0.55 0 0.25 0.09 42 106 188.04 138 138 0.54
0.05 0.19 0.09 57 112 175.39 138 138 0.61
0.5 0.65 0 0.26 0.10 41 104 184.05 135 135 0.54
0.05 0.20 0.09 57 110 175.37 135 135 0.62
0.6 0.75 0 0.27 0.10 37 93 165.29 121 121 0.54 0.05 0.20 0.09 51 98 157.95 121 121 0.62
0.7 0.85 0 0.28 0.10 30 74 132.75 96 96 0.54 0.05 0.22 0.10 41 77 127.27 96 96 0.61
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly. n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 3. Designs for ri r0 0.20, K 2,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.25 0 0.23 0.09 13 33 93.41 51 51 0.45
0.05 0.19 0.09 15 33 74.22 51 51 0.47
0.1 0.3 0 0.28 0.11 17 45 125.25 63 63 0.49
0.05 0.23 0.10 20 45 105.11 63 63 0.52
0.2 0.4 0 0.33 0.12 22 62 168.13 83 83 0.51 0.05 0.27 0.12 28 61 149.63 83 83 0.54
0.3 0.5 0 0.35 0.13 26 71 195.08 95 95 0.51 0.05 0.29 0.13 32 71 174.93 95 95 0.54
0.4 0.6 0 0.36 0.14 27 74 203.00 99 99 0.51
0.05 0.30 0.13 34 74 185.77 99 99 0.55
0.5 0.7 0 0.38 0.14 26 71 195.07 95 95 0.51
0.05 0.32 0.14 33 71 180.69 95 95 0.55
0.6 0.8 0 0.40 0.15 22 62 168.10 83 83 0.51 0.05 0.33 0.14 29 61 158.52 83 83 0.54
0.7 0.9 0 0.43 0.16 17 47 128.50 63 63 0.51 0.05 0.36 0.15 22 46 121.45 63 63 0.54
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly. n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 4. Designs for ri r0 0.15, K 2,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.2 0 0.19 0.07 20 53 147.27 77 77 0.47
0.05 0.14 0.07 25 56 113.56 77 77 0.53
0.1 0.25 0 0.22 0.08 27 75 204.51 101 101 0.50
0.05 0.17 0.08 36 78 170.61 101 101 0.56
0.2 0.35 0 0.25 0.09 38 106 288.59 140 140 0.51 0.05 0.19 0.09 51 109 249.46 140 140 0.57
0.3 0.45 0 0.26 0.10 45 125 340.89 164 164 0.52 0.05 0.20 0.09 61 129 301.42 164 164 0.58
0.4 0.55 0 0.27 0.10 48 134 364.72 174 174 0.52
0.05 0.21 0.10 65 137 323.90 174 174 0.58
0.5 0.65 0 0.28 0.10 47 131 356.78 171 171 0.52
0.05 0.22 0.10 64 134 320.58 171 171 0.58
0.6 0.75 0 0.29 0.11 42 118 320.35 154 154 0.52 0.05 0.22 0.10 58 120 291.48 154 154 0.58
0.7 0.85 0 0.30 0.11 34 93 255.35 122 122 0.52 0.05 0.24 0.11 46 94 232.87 122 122 0.57
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly. n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 5. Designs for ri r0 0.20, K 3,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.25 0 0.24 0.09 14 38 140.80 57 57 0.46
0.05 0.20 0.09 16 38 109.62 57 57 0.47
0.1 0.3 0 0.29 0.11 17 52 183.79 71 71 0.49
0.05 0.24 0.11 21 51 153.52 71 71 0.51
0.2 0.4 0 0.34 0.12 23 71 250.18 93 93 0.51 0.05 0.28 0.12 29 70 218.58 93 93 0.53
0.3 0.5 0 0.36 0.13 27 81 288.60 106 106 0.51 0.05 0.30 0.13 34 80 256.67 106 106 0.54
0.4 0.6 0 0.38 0.14 28 85 301.47 111 111 0.51
0.05 0.32 0.14 35 84 269.53 111 111 0.54
0.5 0.7 0 0.40 0.14 27 82 290.83 106 106 0.51
0.05 0.33 0.14 34 80 261.34 106 106 0.54
0.6 0.8 0 0.42 0.15 23 71 250.16 93 93 0.51 0.05 0.35 0.15 30 70 231.67 93 93 0.54
0.7 0.9 0 0.45 0.16 18 53 190.27 71 71 0.50 0.05 0.38 0.16 23 52 177.16 71 71 0.53
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly. n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 6. Designs for ri r0 0.15, K 3,
K,
0.05,0.2
r0 ri C1 C2 C3 n2 n3 EN n2 n3 Ratio 0.05 0.2 0 0.19 0.07 21 61 219.82 86 86 0.48
0.05 0.15 0.07 27 63 166.69 86 86 0.52
0.1 0.25 0 0.23 0.08 28 86 303.46 114 114 0.50
0.05 0.18 0.08 37 88 244.94 114 114 0.55
0.2 0.35 0 0.26 0.10 39 121 425.32 157 157 0.51 0.05 0.20 0.09 53 123 360.73 157 157 0.56
0.3 0.45 0 0.27 0.10 47 143 506.58 184 184 0.52 0.05 0.21 0.10 63 145 434.03 184 184 0.57
0.4 0.55 0 0.28 0.10 50 153 540.84 196 196 0.52
0.05 0.22 0.10 68 155 470.83 196 196 0.57
0.5 0.65 0 0.29 0.11 49 150 530.18 192 192 0.52
0.05 0.23 0.10 67 151 465.47 192 192 0.57
0.6 0.75 0 0.30 0.11 44 134 474.58 172 172 0.52 0.05 0.24 0.11 60 135 420.49 172 172 0.57
0.7 0.85 0 0.32 0.12 35 106 376.27 137 137 0.51 0.05 0.25 0.11 48 106 337.80 137 137 0.56
n2 ans n3 is the sample size per group of our proposed design at the phase II and III stage correspondingly. n2
ans n3 is the sample size per group of a traditional one-stage design at the phase II and III stage respectively.
Table 7. The result of simulation for the case of ri r0 0.20 , K 1 ,
K,
0.05,0.2
, C10r0 ri Program_alpha Program_power Simulation_alpha Simulation_power
0.05 0.25 0.0505 0.7977 0.0827 0.9292
0.1 0.3 0.0507 0.7932 0.0646 0.8899
0.2 0.4 0.0505 0.7975 0.0557 0.8507
0.3 0.5 0.0505 0.7952 0.0512 0.8122
0.4 0.6 0.0505 0.7961 0.0489 0.7984
0.5 0.7 0.0499 0.7996 0.0492 0.7940
0.6 0.8 0.0499 0.8000 0.0496 0.7808
0.7 0.9 0.0500 0.8004 0.0511 0.7582
Table 8. The result of simulation for the case of ri r0 0.20 , 1K ,
K,
0.05,0.2
, C1 0.05, C2:two-sidedr0 ri Program_alpha Program_power Simulation_alpha Simulation_power
0.05 0.25 0.0507 0.7966 0.0889 0.9287
0.1 0.3 0.0494 0.8020 0.0713 0.8824
0.2 0.4 0.0502 0.7995 0.0551 0.7966
0.3 0.5 0.0501 0.7980 0.0439 0.7924
0.4 0.6 0.0505 0.7964 0.0507 0.7991
0.5 0.7 0.0505 0.7975 0.0417 0.7610
0.6 0.8 0.0500 0.7995 0.0475 0.7825
0.7 0.9 0.0500 0.7987 0.0595 0.7856
Table 9. The result of simulation for the case of ri r0 0.15 , K 1 ,
K,
0.05,0.2
, C10r0 ri Program_alpha Program_power Simulation_alpha Simulation_power
0.05 0.2 0.0505 0.7977 0.0305 0.8786
0.1 0.25 0.0507 0.7932 0.0480 0.8641
0.2 0.35 0.0505 0.7975 0.0474 0.8203
0.3 0.45 0.0505 0.7952 0.0482 0.8255
0.4 0.55 0.0505 0.7961 0.0532 0.8095
0.5 0.65 0.0499 0.7996 0.0503 0.7823
0.6 0.75 0.0499 0.8000 0.0497 0.7744
0.7 0.85 0.0500 0.8004 0.0553 0.7889
Table 10. The result of simulation for the case of ri r0 0.15 , 1K ,
K,
0.05,0.2
, C1 0.05r0 ri Program_alpha Program_power Simulation_alpha Simulation_power
0.05 0.2 0.0508 0.7974 0.0485 0.7910
0.1 0.25 0.0505 0.7975 0.0581 0.8382
0.2 0.35 0.0501 0.7983 0.0550 0.8054
0.3 0.45 0.0506 0.7958 0.0616 0.8209
0.4 0.55 0.0505 0.7960 0.0480 0.7996
0.5 0.65 0.0500 0.7986 0.0462 0.7916
0.6 0.75 0.0503 0.7971 0.0523 0.7974
Figures
Figure 1. Simulated success rates for the case of ri r0 0.20 , K 1 ,
K,
0.05,0.2
, C10Figure 2. Simulated success rates for the case of ri r0 0.20 , K 1 ,
K,
0.05,0.2
, C1 0.05Figure 3. Simulated success rates for the case of ri r0 0.15 , K 1 ,
K,
0.05,0.2
, C10Figure 4. Simulated success rates for the case of ri r0 0.15 , K 1 ,
K,
0.05,0.2
, C1 0.05Appendix
According to the assumption that the estimates of response rates at the phase II stage of the dose groups and the control group are mutually independent and the result of Central Limit Theorem, (rˆ0II,rˆ1II,...,rˆKII)T follows a multivariate normal
we can conclude that
,because of the property of the multivariate normal distribution, where
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